Effects of feedback-free starburst galaxies on the 21-cm signal and reionization history

Different star-formation models at Cosmic Dawn produce detectable signatures in the observables of upcoming 21-cm experiments. In this work, we consider the physical scenario of feedback-free starbursts (FFB), according to which the star-formation efficiency (SFE) is enhanced in sufficiently massive halos at early enough times, thus explaining the indication from the James Webb Space Telescope for an excess of bright galaxies at $z \geq 10$. We model the contribution of FFBs to popII SFE and compute the impact these have on the 21-cm global signal and power spectrum. We show that FFBs affect the evolution of the brightness temperature and the 21-cm power spectrum, but they only have a limited effect on the neutral hydrogen fraction. We investigate how the observables are affected by changes in the underlying star formation model and by contribution from popIII stars. Finally, we forecast the capability of next-generation Hydrogen Epoch of Reionization Array (HERA) to detect the existence of FFB galaxies via power spectrum measurements. Our results show the possibility of a significant detection, provided that popII stars are the main drivers of lowering the spin temperature. Efficient popIII star formation will make the detection more challenging.


INTRODUCTION
Between recombination and reionization, the Universe was permeated with neutral hydrogen.The absorption of CMB photons, collisions between particles and radiation from the first stars excited some of the hydrogen atoms from the singlet to the triplet state.These processes, together with hyperfine transitions that brought the atoms back to the singlet state, sourced the cosmological 21-cm signal (see e.g., Refs.(Furlanetto et al. 2006;Pritchard & Loeb 2012) for review).Its redshift evolution can be used to probe the conditions of the gas in the intergalactic medium (IGM) across cosmic time.
While at very high redshift, in the so called Dark Ages, the 21-cm signal provides direct access to fluctuations in the matter density field, at Cosmic Dawn below  ∼ 30 it becomes particularly sensitive to astrophysical processes related to the formation of the first stars and galaxies.The way neutral HI gas heats and ionizes in this phase depends on the efficiency of star formation and it develops inhomogeneously (Mesinger et al. 2011;Muñoz et al. 2022).Different star formation scenarios may lead to very different reionization histories, which can potentially be probed by next generation 21cm experiments targeting the global signal (e.g., EDGES (Bowman et al. 2018) and SARAS (Patra et al. 2015)) or interferometers measuring the power spectrum, such as HERA (DeBoer et al. 2017), MeerKAT (Wang et al. 2021) and SKA (Ghara et al. 2017).
At the center of our work, we consider instead the scenario of feedback-free starbursts (FFB), which was proposed by the authors of Dekel et al. (2023).Under the conditions of high density and low metallicity expected at high redshifts in massive dark-matter (DM) halos, star formation is predicted to be enhanced by an increased efficiency in converting the accreted gas into stars.At other epochs and DM halo masses, stellar feedback -namely supernovae, stellar winds, radiative pressure and photo-heating-lead to lower efficiency.FFBs naturally emerge when the free-fall timescale for star formation is ∼ O (1 Myr), i.e., shorter than the time required for a starburst to generate effective stellar feedback.Li et al. (2023) further developed the FFB model, showing how it affects different observables during Cosmic Dawn, and simulations are currently under development, aimed at refining the theoretical detail of this scenario.While giving rise to a high abundance of bright galaxies at  ≳ 10, the FFB scenario may also leave imprints on the 21-cm signal and reionization process.
The synergy with 21-cm surveys, therefore, can be the key to provide observable tests for the FFB scenario.The goal of this work is to investigate the effect of FFB galaxies in this context.In section 2 we summarize the modelling required: we introduce the observables (2.1), and how they depend on star formation (2.2) in the standard and FFB scenarios.Section 3 describes the HERA survey characteristics and the setup of our analysis.In section 4 we investigate how FFBs affect the 21-cm observables, namely the global signal and power spectrum.We account for contributions from population III stars in section 5. Finally, we discuss how this analysis translates to constraints on the detectability of the FFB scenario, which we forecast for HERA 21-cm power spectrum, in section 6.We draw conclusions in section 7.

MODEL
To perform our analysis, we need first of all to introduce the 21-cm observables and to characterize how they depend on the underlying cosmological and astrophysical models.Crucial in this sense are the role of star formation and its efficiency; therefore, we summarize the main features of its model in the standard and FFB scenarios.

21-cm observables
The main observables used to analyze 21-cm surveys are the brightness temperature (Barkana & Loeb 2001) and its fluctuations   .In the previous equation,   ∝ (1 + ) is the CMB temperature and is the 21-cm optical depth, which depends on the matter fluctuations , the fraction of neutral hydrogen  HI and the comoving gradient of the baryon peculiar velocity along the line-of-sight     .The dependence on the cosmological model is collected into the Hubble parameter  () and the normalization factor Cosmological parameters {ℎ, Ω  , Ω  ,   ,   } are set at the Planck 2018 (Aghanim & et al. 2020) fiducial values throughout this work.The last ingredient in Eq. ( 1) is the spin temperature   , that quantifies the ratio between the number density of hydrogen atoms in the triplet and singlet states.At thermal equilibrium, the spin temperature is set by where   is the gas kinetic temperature and   ∼   (Field 1959) is the color temperature of the Ly photons emitted by the surrounding stars.The coefficient   ≃ 1 couples the spin temperature to the CMB, while   ,   couple it to the gas temperature.Following Mesinger et al. (2011),   depends on particle collisions and it can be estimated as a function of the number densities of neutral hydrogen, free electrons and free protons; its effect is relevant in the IGM only at  ≳ 30 (Loeb & Zaldarriaga 2004).On the other hand,   is set through the Wouthuysen-Field process (Wouthuysen 1952;Field 1958;Hirata 2006) to be proportional to   (x, )/(1 + ), namely to the Ly background flux due to the integrated star formation rate.The value of   depends on HI excitation due to Xrays (Pritchard & Furlanetto 2007) and to resonant scatterings in the Ly series (Barkana & Loeb 2005).Near the sources, the HI optical depth and the contribution of high energy photons redshifted into the Ly band make this coupling highly efficient.On the other hand, X-rays heat the gas faster; their luminosity is parametrized through a power-law, with a low-energy cut-off below which photons are absorbed before reaching the IGM (Fragos et al. 2013).
The Ly (UV) radiation produced by astrophysical processes also leads to HI ionization.Initially, the process is balanced by the recombination rate (Sobacchi & Mesinger 2014;Park et al. 2019), which stalls the growth of the ionized regions.Once the number of ionizing photons becomes high enough to saturate the Ly coupling and make the interstellar medium transparent to other ionizing photons, these escape into the IGM (Verhamme et al. 2015) and the fraction of neutral hydrogen  HI in Eq. ( 1) decreases, leading to a decay in the 21-cm signal.
All these processes arise inhomogeneously in the IGM: the amplitude and size of local fluctuations determine the 21-cm power spectrum, which is defined as During the epoch of star formation, Ly photons initially couple the spin temperature to the adiabatically-decreasing kinetic temperature.Only after this coupling saturates, Ly photons can heat the gas: this happens earlier in small DM halos, therefore small scales in the power spectrum have larger amplitude in this stage.Large-scale power rises later, but it quickly overcomes the small scales due to X-ray heating, whose efficiency is larger close to individual sources, which are apart one from another.Once X-ray radiation reaches the IGM, fluctuations in the power spectrum are determined by the DM density field and, as time passes, by the morphology of HI ionized regions.Since ionization initially occurs due to UV radiation inside small DM halos (Wood & Loeb 2000), small scale power decreases faster.Once reionization is complete, the 21-cm signal disappears.

Star Formation
Star formation modelling is the key to understanding the 21-cm signal evolution at  ≲ 30.Following Muñoz et al. (2022) (MUN21), we consider a standard scenario in which reionization is driven by atomic cooling galaxies (ACGs) hosting population II (popII) stars, in agreement with faint galaxy observations and the UV luminosity function (Park et al. 2019;Behroozi & Silk 2015;Yung et al. 2021;Fialkov et al. 2013).We then summarize the feedback-free starburst scenario (Dekel et al. 2023) (DEK23) and describe how it alters the star formation rate (SFR) and efficiency (SFE).
In both scenarios, we adopt the formalism from Refs.(Dekel et al. 2013(Dekel et al. , 2023)), that characterizes the SFR per halo as where  acc (,  ℎ ) is the mean baryonic accretion rate,  (,  ℎ ) is the star formation efficiency, and  duty = exp (− turn / ℎ ) includes a turnover mass  turn to suppress the SFR on the small mass end.We approximate  acc using the analytical prescription in DEK23, The SFR in Eq. ( 6) differs from the approximated SFR model in MUN21 and relies on more informed galaxy formation studies (e.g., Refs. (Dekel et al. 2013(Dekel et al. , 2023;;Mason et al. 2015;Tacchella et al. 2018;Mirocha et al. 2021)); in Appendix A, we discuss in detail the difference between the two and their impact on 21-cm observables.
In our analysis, we weight the SFR by the halo mass function 1 / ℎ , and we marginalize over  ℎ to get the SFR density The SFRD is the main quantity that enters the computation of the Ly background and X-ray heating in the 21-cm signal; the shape of the halo mass function implies that the contribution of the more massive halos is suppressed compared to the small mass ones.Moreover, the SFRD enters the computation of the number of ionizing photons.As Park et al. (2019) describes in detail, to compute it we need to introduce the parameter that describes the fraction of Ly ionizing photons capable of leaving the galaxies and ionizing the intergalactic medium; we use  esc = 10 −1.35 ,  esc = −0.3.The value of fesc in the Epoch of Reionization is still uncertain; recent results from CEERS (Noh & McQuinn 2023) seem to point to a mismatch between data and theoretical prescriptions.
It is interesting at this point to note that, while the ionizing fraction depends on fesc , the heating is unaffected by its value (Park et al. 2019).This is due to the longer mean free path that both soft-UV and X-ray photons that heat the gas have with respect to UV photons that drive reionization (Mesinger et al. 2013;McQuinn 2012).In fact, the cross section for the absorption of ionizing photons with energy ≥ 13.6 eV is very high: whenever the HI column density is large, they get trapped inside galaxies and are not capable of reaching the IGM.The recombination of HI atoms inside the galaxies then produces a Ly cascade (Santos 2004) that adds to the bulk of soft-UV photons.Because of their lower energy, these can be absorbed only if the energy matches one of the lines in the Lyman series; the cross section of this process is smaller and results in a longer mean free path, that allows them to reach the IGM.Photons with energy > 10.2 eV are later on redshifted into the Ly line, and interact with HI and diffuse in the IGM as a result of scattering due to their absorption and re-emission (Santos 2004).
The main consequence of the different mean free paths of ionizingand heating-photons is their dependence on the distribution of HI column density regions (Das et al. 2017).While the former is described by Eq. ( 9), which leads to a suppression of the ionization in the more massive halos, the latter depends on the formation efficiency of the sources that mainly produce the radiation field.In our analysis, we consider three main drivers: popII stars, formed in atomic cooling galaxies; FFBs; popIII stars formed in molecular cooling galaxies.We characterize popII and FFB efficiency in the next subsections, while popIII stars are investigated in Sec. 5.

Atomic Cooling Galaxies
In the ACG scenario where popII stars are formed, we use the prescription in MUN21 to characterize the SFE in the standard case  (,  ℎ ) =  MUN21 ( ℎ ), in which 1 We adopt the -dependent halo mass function from (Watson et al. 2013).

Feedback-Free Starburst Galaxies
When modelling star formation, the role played by stellar feedbacks, such as winds or supernova explosions, is crucial.The star formation efficiency at low  is believed to be small due to feedback (Rodríguez-Puebla et al. 2017;Moster et al. 2018;Behroozi et al. 2013Behroozi et al. , 2019)), while the FFB scenario introduced by DEK23 shows that the SFE is higher in massive galaxies at  ∼ 10, in agreement with the excess of bright galaxies in JWST observations.
For FFB to happen, the free-fall collapse time of the star-forming cloud (SFC) have to be shorter than the time required by the stellar feedbacks to become effective.The former is estimated as where  SFC is the gas number density, while the latter is  fbk ≃ 1 Myr.Moreover, the timescale  ff has to be larger than the time the gas requires to cool and form stars,  cool ∝  −1 SFC (Krumholz 2012).Finally, a large enough surface density Σ SFC =  SFC / 2 SFC is required to prevent unbounding the SFC gas through stellar radiative pressure and photo-ionization,  SFC ,  SFC being its mass and radius (Fall et al. 2010;Grudić & et al. 2018;Grudić et al. 2020).In order for these processes to realize efficiently, not only do SFCs have to be free of their own feedbacks, but they also need to be shielded against UV radiation and winds from older-generation stars.
The former is guaranteed, since  SFC is larger than the ionizing length  inside which the UV photon flux overcomes the recombination rate.As for shielding against stellar winds (see e.g., Menon et al. (2023)), the time a shock wave takes to cross the SFCs, namely the cloud crushing time (Klein et al. 1994) SFC , has to be longer than the timescales  ff ,  cool , previously introduced.Gas inside SFCs where FFB take place is almost completely consumed, so they reach near-zero column density (Menon et al. 2023).It is possible that the evolution of the gas temperature, which is larger at higher , has a non negligible impact on the onset of FFBs.However, its effect is not straightforward and it requires the use of numerical simulations to be understood, e.g., to model the way it alters the stellar mass function resulting from the fragmentation process.We defer to a future, dedicated work the study of this effect, while in this paper we rely on the simplified assumptions in DEK23.All these conditions are satisfied only by short starbursts in DM halos continuously supplied by gas, all of which fragment into SFCs.10)) and when FFBs from DEK23 are included (bottom, Eq. ( 14)) as a function of the halo mass and redshift.The white line indicates the threshold mass  FFB for FFBs from Eq. ( 11).Right panels: SFRD from Eq. ( 8) marginalized over  ℎ in the standard case using different  * values (top) and adding FFBs with different  max (bottom).The fiducial value of  * in both the panels is 10 −1.48 .FFBs affect star formation differently from just rescaling the efficiency.
As DEK23 shows, the criteria for the onset of FFB can be translated in terms of the properties of the host DM halo mass at given .Star formation in the halo is driven by FFB, thus its efficiency is enhanced with respect to the standard scenario when This threshold has been computed in DEK23 and it is shown in Fig. 1 Star formation inside halos that satisfy the condition in Eq. ( 11) does not proceed with constant rate.As Li et al. (2023) describes in detail, during the FFB phase an halo undergoes multiple bursts of extremely high star formation, each of which consumes almost all the gas available in the star-forming clouds.The burst is hence followed by a period of quenched star formation, during which new gas accretes onto the galaxy, so to reach high enough density to fragment again.While each burst lasts a few Myr, namely a few times the freefall timescale of the star-forming clouds, the interval between two subsequent bursts is ∼ 10 Myr.Overall, the conditions for FFBs can be materialized over a global period of ∼ 100 Myr, namely the time required to accrete ∼ 10 10  ⊙ of gas, during which ten bursts can be realized, each leading to the formation of one generation of stars.DEK23 already showed that Eq. ( 11) ensures that each generation is shielded against feedback from the previous one.Following the approach in Li et al. (2023), in the following we approximate the SFE during this time interval using a constant value, which averages the rate between the bursts and the times between them; this average quantity turns out to be larger than the SFR in the standard scenario.
Eq. ( 11) highlights the fact that halos of mass ∼ 10 10.8  ⊙ at  ∼ 10 can host FFBs; the threshold decreases at higher redshift, where their presence significantly affects star formation.At low , instead, the threshold mass gets larger, but at the same time the onset of AGN feedback and the presence of hot circumgalactic medium quench star formation in halos  ℎ ≥  q = 10 12  ⊙ .Thus, in the local Universe, FFBs are unlikely.
The way FFBs contribute to the total star formation rate per halo SFR tot (,  ℎ ), can be modelled as where SFR FFB =  max  acc , and the parameter  max ≤ 1 describes the maximum SFE that FFB galaxies can reach.The (,  ℎ ) dependence, which we left implicit for brevity, is encoded in where F ≤ 1 is the fraction of galaxies that form in halos with  ℎ >  FFB () and host FFBs, while S [] = (1 + e −  ) −1 is a sigmoid function varying smoothly from 0 to 1.The first sigmoid characterizes the quenching for  ℎ ≥  q , while the second sets the star formation rate to its value in the standard model SFR std for halos below the threshold  FFB (), while it gains a SFR FFB contribution in halos that host FFBs.
The relation in Eq. ( 12) can be translated to a relation between the SFE in the standard case ( MUN21 , from Eq. ( 10)) and the SFE in galaxies where star formation is driven by FFBs.We consider3 where we kept the  * normalization for consistency.We set { max , F } = {1, 1} as fiducial values, meaning that all the galaxies formed in halos with  ℎ ≥  FFB () have SFE up to  max = 1.In the analysis, we keep F = 1 fixed, but we test the more conservative case  max = 0.2, where FFB galaxies reach smaller efficiency.
The left panels of Fig. 1 compare the standard SFE  MUN21 ( ℎ ) from Eq. ( 10) with  tot (,  ℎ ) from Eq. ( 14), in which FFBs with  max = 1 are included.In the right panels, the figure compares the standard SFRD with the cases of interest for our analysis.Finally, in Appendix C we show how the high- UV luminosity function changes when FFBs are included.The figure can be compared with Fig. 5 in Li et al. (2023), where a more detailed discussion on this observable can be found.

ANALYSIS SETUP
The enhanced star formation efficiency from Eq. ( 14) naturally has a non-negligible impact on the 21-cm signal.
To estimate the effect of FFBs on 21-cm obervables, we customized the public code 21cmFAST4 (Mesinger et al. 2011;Muñoz et al. 2022;Murray et al. 2020).The code simulates the reionization history by modelling the radiation fields we described in Sec.2.1 and their effects on the thermal evolution and neutral hydrogen fraction inside cells of the simulation box.The evolution of cosmological fluctuations can be consistently accounted for via the initial conditions in 21cmFirstCLASS (Flitter & Kovetz 2023a,b).We modified the ACG SFE by introducing the redshift-dependent  tot from Eq. ( 14); as described in the previous Section and in Li et al. (2023), this quantity is used to approximate with a constant, average value the SFE during the ∼ 100 Myr in which an halo satisfies the condition set in Eq. ( 11) to host FFBs.We changed  tot at each  only for halos close to and above the mass threshold in Eq. ( 11), according to Eq. ( 13).Halos that exit the FFB condition behave as in the standard scenario, with no consequences due to their previous state.This is justified by the fact that, after ∼ 20 Myr from the last burst of star formation, most of the massive stars produced during the FFB phase have evolved and exited their active phase.Therefore, their presence does not increase the amount of feedback with respect to the standard case, and so SFE can be restored to its standard value.
Throughout the analysis, we used a (256 Mpc) 3 simulation box, inside which 384 cells are defined on each axis for the high resolution computation, related with initial condition and displacement field, and 128 for the low resolution one, for temperature and ionization fluctuations (Mesinger & Furlanetto 2007).We used initial adiabatic fluctuations described by the approximation in Muñoz (2023), the CLASS configuration for the matter power spectrum, the -dependent mass function from Watson et al. (2013), and we included the effects of redshift space distortions and relative velocities between dark matter and baryons.The code smooths the density perturbations over neighbouring cells; finally, Ly heating is also included (Sarkar et al. 2022) (while CMB heating is not).
We estimate the global signal and neutral hydrogen fraction based on lightcones produced by 21cmFAST , and compute the 21-cm power spectrum with powerbox5 (Murray 2018).

Noise model
We compute the 21-cm power spectrum noise when observed with HERA, the Hydrogen Epoch of Reionization Array.Forecasted HERA sensitivity depends on the detector configuration and on the goodness of foreground removal.Following Refs.(Parsons et al. 2012;Pober et al. 2013Pober et al. , 2014)), the baseline length between each pair of detectors sets the transverse modes  ⊥ that can be observed, while the bandwidth sets  ∥ along the line-ofsight.Spectral-smooth foregrounds (i.e., Galactic and extragalactic synchrotron and free-free emissions) mainly contaminate small  ∥ .Their contribution, however, is then processed through the chromatic response function of the instrument: the mode-mixing sourced by the interferometer itself can deteriorate the signal on different angular locations as a function of frequency.It has been shown (see e.g., (Datta et al. 2010)), that this effect is mainly relevant for large  ⊥ : this determines a wedge-shape in the ( ⊥ ,  ∥ ) plane.As a result, the values of  ∥ associated with small  ⊥ are mostly foregroundfree.Pober et al. (2014) defines different foreground removal models depending on the shape of the wedge edge: in this work, we will adopt the "moderate" and "optimistic" foreground removal scenarios.The former extends the edge to 0.1 ℎMpc −1 beyond the horizon where | ì | is the baseline length,  the speed of light and  = (1 + ) 2 / 21  () is the conversion factor between bandwidth and line-of-sight distance, while  21 ∼ 1420.4MHz is the rest frame frequency associated with the 21-cm line.The optimistic model, instead, improves the constraining power on both the small and large scales by extending  ∥ to the FWHM of the primary beam, computed as FWHM = 1.06 obs /14 m ∼ 10 • .Both models assume to add coherently different baselines, namely the integration times are summed when the same pixel is sampled more than once by redundant baselines.More detail on the noise computation is given in Refs.(Pober et al. 2013(Pober et al. , 2014)).
To estimate the HERA sensitivity, we rely on the public code 21cmSense7 (Pober et al. 2013(Pober et al. , 2014)), which combines the contribution from the thermal noise power spectrum and the sample variance.Here,  converts the observed angles into transverse measurements,  sys is the system temperature,  the duration of the observational run and Ω = 1.13 FWHM 2 the solid angle associated with the primary beam.The sample variance instead is estimated as the 21-cm power spectrum in Eq. ( 5) and summed with the thermal noise to get the noise variance  2 HERA .In our analysis, we consider a hexagonal configuration with 11 dishes per side (i.e., 331 antennas in total), each having 14m diameter.In Eq. ( 16), the observational time  is set to 6 hours per day over 540 days, while  sys =  sky +  rcv , where the sky temperature is  sky = 60 K/(/300 MHz) 2.55 and the receiver temperature is  rcv = 100 K.We consider a minimum observed frequency of 50 MHz, a maximum frequency of 225 MHz and 8 MHz bandwidths probed by 82 channels each.This sets the observed redshift bins to The 19 bins obtained are equally spaced in frequency but not in redshift, providing a finer sampling at low .

FFB SIGNATURES ON 21-CM OBSERVABLES
The analysis presented in this work has been realized using the public codes: 21cmFAST , version 3.3.1 updated in June 2023; powerbox; 21cmSense .We modified 21cmFAST to include FFB galaxies and to account for the SFR formalism defined in Eq. ( 6).As we discuss in detail in Appendix A, this SFR model differs from the approximation used in 21cmFAST public release: in the standard scenario, it provides a slightly larger SFR, thus anticipating reionization with respect to, e.g., results in MUN21.
We checked that, in the range of scales probed by HERA, using a single realization of the power spectrum or averaging over 5, 10 or 15 21cmFAST simulations provides variations smaller than the error bars.Therefore, to reduce the computational cost, plots and forecasts are realized using the same random seed.

Global signal
First of all, we investigate how FFBs affect the 21-cm global signal.This observable, in fact, can help us understanding in a more straightforward way the peculiarities the FFB scenario has compared with other cases.
Fig. 2 shows how the presence of FFBs impacts the brightness temperature and reionization once different values of  max are considered.The value of   in Eq. ( 1) is estimated for our FFB prescription in Eq. ( 14) and compared with the standard 21cmFAST configuration from Eq. ( 10).For comparison, we also consider an artificial toy model in which  MUN21 is increased over all the halo masses and the entire redshift range, by simply rescaling the value of  * by 3; the SFRD for the same model is also shown in Fig. 1.This value was chosen to match the star formation efficiency required by JWST observations at  ∼ 9 without FFBs (see e.g., Refs.(Labbé et al. 2023; Boylan-Kolchin 2023)), but has no particular physical meaning.
When star formation efficiency is increased at high redshift, a larger amount of Ly and X-ray radiation is produced, speeding up the coupling of the spin temperature to the gas temperature and anticipating the moment in which this heats up.Therefore, in the top panel of Fig. 2, both the 3  * and FFB cases induce a larger   global signal and move its peak towards large .Moreover, the increased Ly flux enhances the efficiency of the coupling between the spin temperature   and the gas temperature   ∼   in Eq. ( 4).Even if the gas at higher redshift is still hot, the stronger coupling drives   farther from the CMB temperature   , leading to a larger difference in the computation of the brightness temperature in Eq. ( 1).For this reason, both in the 3  * and in the FFB models in Fig. 2 the   absorption peak becomes deeper when shifted at higher .However, as more time passes, only the more massive halos keep satisfying the conditions that allow the presence of FFB, while SFR in halos with masses between 10 8  ⊙ and 10 10  ⊙ becomes less efficient.
While one might expect the high efficiency of the FFBs to result in a more efficient reionization, we find that it is not necessarily the case.As described in Sec.2.1, the reionization rate is determined by the escape fraction fesc of the ionizing photons.In the model underlying our assumption, the negative value of  esc , motivated by Lyman- forest and CMB data (Qin et al. 2021), penalizes the contribution of large halos.Since these are indeed the one that host FFBs, the effect of FFBs on  HI is limited.In contrast, if we were using positive values of  esc we would increase the ionizing power of massive halos and thereby improve the relevance of FFBs on reionization.
In Appendix B we show how the brightness temperature evolves inside lightcones produced by 21cmFAST.The behaviour of the temperature in the lightcones and the presence of structures on the small scales reflect the evolution of the global signal and of the power spectrum, which we describe in the next Section.
As a side comment, note that we adopted the same expression for the escape fraction fesc in Eq. ( 9) for both non-FFB and FFB galaxies.This quantity is determined by the neutral hydrogen column density inside a galaxy, which describes the abundance of atoms capable of absorbing the ionizing radiation, preventing it to reach the IGM.In principle, the presence of FFBs may increase fesc , since they consume all the gas in the star-forming clouds, and they remove dust via steady wind (Li et al. 2023).As the escape fraction in the Epoch of Reionization is anyway very uncertain, we leave further investigation on how it gets affected by FFBs to future work.
To sum up, when FFBs are taken into account, the peak in the 21-cm global signal starts at higher , reaches a lower value and then ends slightly before the standard scenario; smaller values of  max or F make the effect less significant in an almost-degenerate way.This is different from what we would expect for an overall increased SFR, as we model with 3  * .Here, the peak shifts at higher  but, thanks to the large efficiency in small mass halos, reionization is faster and the signal reaches   = 0 earlier.

Power spectrum
FFBs also affect the 21-cm power spectrum.Fig. 3 shows its redshift evolution: consistently with the global signal, the ionization bump rises earlier, at  ∼ 8, for the 3  * model, while for FFBs it matches the standard scenario at  ∼ 6.The presence of massive FFB-hosting halos increases Δ 2 21cm at high ; their lack of ionization power would keep the signal amplitude large even at low , but the contribution of small halos brings Δ 2 21cm back to the standard case.The errorbas in the figure are estimated for HERA with moderate foreground, assuming the FFB scenario as fiducial, as in Sec.3.1; qualitatively, FFB signatures on the 21-cm power spectrum seems to be distinguishable in certain (, ) ranges.
Finally, in Fig. 4 we compare the redshift evolution of the scaledependent power spectrum with and without FFBs.These plots were obtained using a 700 Mpc simulation box to access larger scales.At very high redshift, the power spectrum in the FFB scenario has larger power; this can be understood comparing the amplitude of the global signal at the same epoch.At 8 <  < 13, FFBs experience a boost initially on the large, then on the small scales: only the rare, most massive halos can still host FFBs at this redshift, thus increasing correlation on large scales; since these halos are the most densely clustered, they lead to an increase in the small scale power as well.Lower redshifts, instead, are dominated by the contribution of small halos; as already discussed, this brings the shape of the power spectrum back to the standard case.

INCLUDING MOLECULAR COOLING GALAXIES
A further contribution to the SFR and 21-cm signal could come from population III (popIII) stars (Bromm & Larson 2004;Bromm 2005).Usually, popIII stars are associated with a pristine, metal-poor environment, and their formation is driven by H2 molecular cooling (see e.g., (Haiman et al. 1996(Haiman et al. , 1997;;Abel et al. 2002)).As in MUN21, we consider their formation as associated with molecular cooling galaxies (MCG) inside mini-halos, whose typical mass is ∼ 10 7  ⊙ .Their contribution to the reionization process is still uncertain, see e.g.Refs.(Qin et al. 2020;Wise et al. 2012;Xu & et al. 2016), and not yet completely accepted.For example, recent results from HERA Phase I do not account for MCGs in their modelling; including them, depending on their efficiency, can lead to variations in the parameter constraints (Lazare et al. 2023).
In this section, we model popIII contribution to SFE and we study how this affects the 21-cm observables, accounting for uncertainties.We assume MCGs cannot host FFBs, since the modelling in DEK23 refers to atomic cooling SFCs and the threshold mass in Eq. ( 11) penalizes mini-halos, its value being  FFB > 10 7  ⊙ up to  ∼ 40.Thus, in our formalism, FFBs only enhance SFR for popII stars.The impact of different feedback levels in the formation of popIII stars, capable of accounting for all the uncertainties in the modeling of popIII star formation, requires a more detailed investigation.This however is beyond the scope of the current work; we thus limit our analysis by modeling differently popII (with and without FFB effects) and popIII, and dividing our results to the cases where popIII stars are either neglected or modeled in their standard scenario.We postpone to a follow-up paper the analysis of more complete cases, in which popIII stars can form in massive halos, or could be affected by FFBs.

Model
Following MUN21, we approximate SFR in MCGs as 8 where  * = 0.5 is a fudge parameter and   () =  () −1 is the Hubble time.The SFE is estimated as 8 See further discussion on this SFR approximation in Appendix A.
where we set  III * = 0 and we normalize with respect to the SFE in halos with  ℎ = 10 7  ⊙ , namely  III * .9We choose as nominal value 10 −2.5 ; to account for uncertainties in the MCG efficiency, we also test  III * ∈ [10 −1.5 , 10 −3.5 ].We use to suppress star formation on the large halo mass end, where MCG star formation transits into the ACG scenario.The value of  mol ∝   cb  LW accounts for quenching of star formation on the small mass end.In mini-halos, this is caused by the relative velocity between DM and baryons  CB (Tseliakhovich & Hirata 2010;Naoz et al. 2012;Dalal et al. 2010;Tseliakhovich et al. 2011) and by Lyman-Werner feedbacks (Machacek et al. 2001;Kulkarni et al. 2021;Schauer et al. 2021) due to photons with energy between 11.2 and 13.6 eV, that photo-dissociate molecular hydrogen and prevent the cooling of the gas clouds.We fix the relative velocity contribution to where   cb = 1,   cb = 1.8, the rms velocity is  rms =  avg √︁ 3/8 and we set the average velocity to  avg = 25.86 km/s.As for LW feedbacks, we use (Visbal et al. 2014) , where  LW = 2,  LW = 0.6 and  21 is the LW intensity in units of 10 −21 erg s −1 cm −2 Hz −1 sr −1 .The escape fraction from mini-halos is modelled analogously to Eq. ( 9), with  III esc =  esc ,  III esc =  esc and using 10 7  ⊙ as the normalizing mass scale instead of 10 10  ⊙ .

Effect on the 21-cm observables
The presence of MCGs inside mini-halos changes the 21-cm observables, mainly because of the larger radiation produced at high redshift.As Fig. 5 shows in the left panel, in the standard 21cmFAST case with MCGs, the global signal peak broadens and is preponed, leading also to an earlier reionization (although ACGs remain the main driver).Since popIII stars contribute also to the X-ray emission, their presence heats the gas faster, thus the   peak becomes not as deep as in the only-ACG case.Analogously, the power spectrum shown in the right panel of Fig. 5 has larger power at high , while it dies faster because of the earlier reionization.
The relevance of all these effects depends on the MCG star formation efficiency, encapsulated in the parameters  III * and  III * in Eq. ( 19).In particular, following MUN21, we adopt  III * = 0: this choice renders the popIII star formation efficiency independent from the mass of the mini-halos up to the turnover mass.If, instead, we had chosen  III * < 0, star formation in smaller halos would have been accelerated.On the other hand, effects due to large values of  III * partially cover the high- contribution of FFB galaxies in both the observables.It is clear then that accounting for MCGs makes more challenging to detect the signatures of the FFB scenario.

FISHER FORECASTS
In the previous sections, we estimated the effect of the existence of FFB galaxies on the 21-cm global signal and power spectrum.We now want to understand if HERA (DeBoer et al. 2017) will be able to detect the signatures of this scenario, provided the uncertainties on the MCGs contribution described in Sec. 5.For simplicity, we begin this analysis by ignoring the contribution of popIII stars; later, we relax this assumption in Sec.6.2.Uncertainties related with the SFR model are discussed in Appendix A.
In the parameter set,  max describes the properties of the FFB scenario, {log 10  * ,  * } characterize the ACG star-formation efficiency, {log 10  esc ,  esc } the escape fraction and log 10 (  /SFR) the X-ray luminosity.Degeneracies between the parameters are accounted via the process of marginalization; more details on their role in determining the 21-cm signal can be found in Sec.2.1 and MUN21.Fiducial values are summarized in Tab.1; we use uninformative priors on all the parameters.Variances  2 HERA are computed through 21cmSense for the FFB scenario and including thermal noise and sample variance.The sum is performed over the  bins computed by 21cmSense and the 19 -bins defined by HERA 8 MHz bandwidth.

FFB detectability
First of all, we consider only the contribution of ACGs and FFBs, as described in Sec. 2. We estimate that, in the case of moderate foreground, the relative marginalized error on  max = 1 is   max / max ≃ 6%; optimistic foreground improves the result to   max / max ≃ 1%.Provided that the relative difference between  = 10 SFRD in the standard-ACG and FFB scenarios is ∼ O (50%), our analysis shows that the existence of FFBs can be detected with high significance both considering moderate and optimistic foreground removal.This was expected from the qualitative description in Sec.3: FFBs have a relevant impact on the 21-cm power spectrum and unique features with respect to a simple enhancement of SFR.Thus, degeneracies between parameters entering the Fisher computation are tiny; we check that degeneracies are small between FFBs and other ACG-related parameters through the contour plot in Fig. 6.
Smaller values of  max lead to closer SFRD in the two scenarios and to a weaker constraining power in the 21-cm analysis.For example,  max = 0.2 yields   max / max ≃ 10% with moderate foreground and ≃ 1% with optimistic foreground, against a relative difference ∼ O (10%). 10 We note that decreasing the fraction of FFB galaxies in the massive halos via the F parameter in Eq. ( 12) would lead to similar considerations, since its value is degenerate with  max .
This case represents our benchmark, providing the best results we can get assuming the SFR model is known and described by Eq. ( 6). Discussion on SFR model uncertainties can be found in Appendix A.

Effect due to MCG contributions
We now account for contributions from popIII stars hosted by MCGs.To do so, we compute the Fisher matrix including in the parameter set also {  III * ,  III * }; other popIII-related parameters in 10 The absolute error is ∼ 0.02 when  max = 0.2, and ∼ 0.06 when  max = 1.The fact that the constraining power is comparable in the two cases can be explained by the non-trivial way the 21-cm power spectrum evolves as a function of (,  ).For example, from Fig. 3, one can see that in some redshift ranges Δ 2 21cm is larger for  max = 0.2 than for  max = 1.This leads to a larger signal in some of the observed bands, and to the possibility of getting good constrain when  max = 0.2.Moreover, the Fisher matrix in Eq. ( 22) accounts for partial degeneracies with other astrophysical parameters; when  max = 0.2, their effect can be constrained more easily, hence reducing the uncertainties and relatively improving the constraining power on  max .to account for the large uncertainties on this parameter.The "nominal" case assumes log 10  III * = −2.5;"high efficiency MCGs" adopt log 10  III * > −2.5; and finally "low efficiency MCGs" consider log 10  III * < −2.5.We discuss  max = 1 for conciseness; smaller values lead to less stringent constraints, consistently with results discussed in Sec.6.1.Fig. 7 collects our results on  III  max , i.e., the marginalized error on  max once MCGs are included in the analysis.In the case of moderate foreground, the presence of MCGs lowers the significance of the FFB detection: while with "nominal" and "low efficiency MCGs" values, FFB signatures can be still partially detectable, for "high efficiency MCGs" the power spectrum becomes almost indistinguishable from the scenario without FFBs.The situation changes when optimistic foreground is considered: here, FFBs can be detected in both the "nominal" and "low efficiency MCGs" cases, while for "high efficiency MCGs" the FFB detection is still plausible, even if with smaller significance.Even if the conditions for optimistic foreground are quite hard to reach, this result sets a benchmark for HERA's constraining power on FFBs.The two lines in Fig. 7 hence pinpoint the reasonable detection level we will achieve with future full-HERA data-analysis, provided that foreground cleaning algorithms will reach the foreseen accuracy.

CONCLUSIONS
Upcoming years will provide improved measurements of the 21-cm global signal and power spectrum from the Epoch of Reionization.Combined with other probes, 21-cm experiments will shed light on the processes that regulate star formation in the first galaxies.Uncertainties still exist on the star formation modelling, particularly regarding the role of popIII stars and stellar feedbacks, for which observations in the local Universe suggest an important role in quenching star formation efficiency.An extrapolation of feedback models to high redshifts should take into account other phenomena.
The authors of Dekel et al. (2023) introduced the process of feedback-free starbursts, namely star formation events with short timescales that should arise in high redshift galaxies.For these to be efficient, gas clouds in which star formation takes place have to be dense enough and with low metallicity; these conditions guarantee that star formation has enough time to be realized before stellar feedbacks become effective.Moreover, under similar conditions, starforming clouds would be shielded against radiation and winds from older stars.Overall, it is possible to show that these processes boost star formation efficiency inside halos above a certain mass threshold, whose value increases with cosmological time.Therefore, in the late Universe, feedback-free starbursts are rare since they can only be hosted by very massive halos; moreover, once AGN feedbacks set up, star formation always gets quenched in halos > 10 12  ⊙ .On the contrary, at high redshift the evolution of the threshold mass indicates that feedback-free starbursts can be found even in smaller halos; their presence could explain the existence of high redshift, massive galaxies observed by JWST.It is hence important to analyse further implications this model can have, so to understand which are the observable signatures that could either confirm it or rule it out.
Our work, together with Li et al. (2023), represents one of the first steps in this direction.We investigated the observational signatures of the feedback-free starbursts on the 21-cm signal.We modelled their contribution to star formation efficiency in atomic cooling galaxies and implemented it into 21cmFAST to estimate their effect on the 21-cm global signal and 21-cm power spectrum.
Our main results can be summarized as follows.
• The redshift and mass dependence of the SFE in the FFB scenario speed up the evolution of the brightness temperature and of the 21-cm power spectrum before  ∼ 15.At lower redshift, instead, their evolution gets closer to the non-FFB scenario.These result respectively from the coupling between the spin and gas temperatures, and from the X-ray heating: the coupling is stronger at high  when FFBs are accounted for, due to the low-mass halos that host FFB galaxies at those times; the heating, instead, gets effective at lower , where only massive halos can still host FFBs.
• On the other hand, the evolution of the neutral hydrogen fraction is only weakly affected by the presence of feedback-free starbursts.This is because the low-mass halos with high escape fraction of ionized photons host FFBs only prior to  ∼ 15, practically before the onset of reionization.At lower redshift, such halos tend to be without FFBs, and they therefore contribute to reionization similarly to the standard scenario.On the other hand, the high-mass FFB galaxies at these later times have a negligible contribution to reionization because of their lower escape fraction.
• We forecasted the detectability of the FFB scenario in the different regimes.We showed that future interferometers, such as HERA, will be able to detect signatures of their existence in the 21-cm power spectrum, compared with the standard scenario that only includes popII stars formed in atomic cooling galaxies.We also checked how our results change when the FFB efficiency is lower.
• We accounted for the possible contribution at high redshift of popIII stars in molecular cooling galaxies and showed that this may hide the effect of FFBs.We drew forecasts as a function of popIII efficiency: our results show that, except for cases with high efficient popIII star formation, signatures of the FFB scenario can still be detected.The significance level will depend on the foreground level.
To conclude, our work highlights the crucial role 21-cm experiments can have in testing astrophysical scenarios.Their synergy with other probes, such as JWST data, in the upcoming years will foster our research of the high redshift Universe, helping us to shed light on the puzzles related to reionization and the birth of the first galaxies.

A2 FFB constraints
As a check to the reliability of our analysis, we apply the approximated SFR formalism to the study of FFB detectability.This allows a more straightforward comparison with other works in the 21-cm literature, which adopt the same prescription, e.g., MUN21.Analogously to Sec. 6.1, we run the Fisher analysis using the approximated SFR model, with the same parameter set .For conciseness, we only discuss  max = 1, but a similar analysis can be performed for other values.Since the relative difference between the standard and FFB scenarios is not changed by the change in the SFR model, results using the approximated SFR depart from Sec. 6.1 ∼ O (1%) both un-der moderate and optimistic foreground assumptions.Constraints in the approximated model slightly improve since the power spectrum moves to lower , where HERA is more sensitive.
As Fig. A1 highlights, the difference between the nominal and approximated formalism can be reproduced in the 21-cm observables by varying the value of  * ; larger values of  could also be used to match the two models.Thus, the degeneracy between the SFR model choice and the FFB existence can be at first order understood in terms of the degeneracy between  * ,  and  max .This is represented by the ellipses in Fig. 6 and accounted for using the marginalization in the Fisher matrix computation: given our results, HERA should be able to disentangle features in the 21-cm power spectrum due to the presence of FFBs from uncertainties in the SFR model.

APPENDIX B: LIGHTCONES
In Fig. A4 we show the lightcone evolution of the brightness temperature in one slice of the boxes produced by 21cmFAST.As in the main text in Sec. 4, we show here four different models: the "Standard" described by Eq. ( 10), also in the case of large  * , and the cases in which FFB are included using Eq. ( 14), with  max = {0.2,1}.For comparison, we also show the lightcone in the standard case, when MCGs are included (see Sec. 5).
The behaviour of the temperature in the lightcones reflects the evolution of the global signal in Figs. 2 and 5: regions that are more yellow indicate where the peak is deeper, hence clearly the presence of FFB anticipates and deepens it.Also, it is possible to note how the presence of FFB does not strongly alter the reionization epoch.Finally, one can interpret the evolution of the 21-cm power spectrum described in Sec. 4 by looking at how structures on the different scales form and evolve inside the lightcone.

APPENDIX C: UV LUMINOSITY FUNCTION
Through 21cmFAST, we estimate the luminosity function of high- galaxies in the scenarios in which FFBs are or are not included.The luminosity function Φ( UV , ) is defined as the number density of galaxies per UV magnitude bin, where the absolute magnitude relates to the luminosity  UV through (Oke & Gunn 1983) log 10  UV erg −1 s −1 Hz −1 = 0.4 × (51.63 −  UV ).The -axis shows the rest frame UV absolute magnitude.We compare our results with a compilation of results from JWST (Leung et al. 2023;Harikane et al. 2023;Finkelstein et al. 2023), including lensed fields (Willott et al. 2024).
where the conversion from halo mass  ℎ to luminosity  UV is performed knowing that  UV ( ℎ , ) = SFR( ℎ , ) 1.5 × 10 −28  ⊙ yr −1 . (C3) The luminosity functions of the models analysed in the main text are shown in Fig. C1, compared with a selection of data from HST and JWST.An extended discussion on the luminosity function in the presence of FFBs can be found in Li et al. (2023).

Figure 1 .
Figure 1.Star formation models.-Left panels: SFE in the standard MUN21 case (top, Eq. (10)) and when FFBs from DEK23 are included (bottom, Eq. (14)) as a function of the halo mass and redshift.The white line indicates the threshold mass  FFB for FFBs from Eq. (11).Right panels: SFRD from Eq. (8) marginalized over  ℎ in the standard case using different  * values (top) and adding FFBs with different  max (bottom).The fiducial value of  * in both the panels is 10 −1.48 .FFBs affect star formation differently from just rescaling the efficiency.

Figure 2 .
Figure 2. FFB effect on   and  HI .-  global signal (left) and neutral hydrogen fraction  HI (right) in the standard scenario using either the nominal  * (black) or 3  * (gray, dotted), compared with the case that includes FFBs with  max = 1 (orange) or 0.2 (magenta, dashed).FFBs anticipate the   peak, while they have negligible effect on  HI , due to low fesc in massive halos.

Figure 4 .
Figure 4. FFB effect on Δ 2 21cm redshift evolution.-Powerspectrum in the standard scenario (black continuous and dotted lines, respectively  * and 3  * ) and including FFBs (orange,  max = 1).The orange shaded area shows  HERA , while the gray area indicates the −range not probed by HERA.Here we run 21cmFAST in a larger, 700 Mpc side box, to understand how FFBs contributes to larger scales.At  = 6.3, there is no 21-cm power spectrum for 3  * since reionization is complete.FFBs boost large and then small scales at 8 <  < 13, since they form in the most massive and more clustered halos.

Figure 6 .
Figure 6.Confidence ellipses.-Marginalized1 ellipses in the ACG+FFB case from Sec. 6.1 when  max = 1.The FFB parameter  max has small degeneracies with other ACG-related parameters that affect the power spectrum.

Figure 7 .
Figure 7. Summary of our constraints on FFBs.-Marginalized 1 error on  max including MCG as a function of log 10  III* , with moderate (orange) and optimistic (magenta) foreground.Horizontal dashed lines mark the case with only ACGs described in Sec.6.1.The black line shows  max / max = 1/3 as a reference, for which  max can be detected ∼ O (3).FFB signatures can be detected by HERA when popIII star formation efficiency is not too high.

Figure A2 .
Figure A2.  and  HI using different SFRs.-Global signal (left) and neutral hydrogen fraction (right).Solid lines use the nominal SFR, while dashed the approximated SFR.The magenta line shows the nominal case adopted in our analysis, where log 10  * = -1.48so that the nominal SFR agrees with the approximated one.The black, continuous line instead shows the nominal model when  * has the same fiducial value as MUN21.The difference between the nominal and approximation models is captured by  * .

Figure
Figure A3.Δ 2 21cm using different SFRs.-Power spectrum at large (left) and small (right) scales as a function of .Same legend as in Fig. A2.The shaded area shows ± HERA with moderate foreground with respect to the nominal SFR model in the standard scenario (no FFBs).The difference between the nominal and approximation models is captured by  * .

Figure A4 .
Figure A4.Lightcones produced by 21cmFAST for the different models analysed in the main text (see Sec. 4. The colorbar indicates the value of the 21-cm global signal, while the -axis shows the redshift evolution (note that smaller redshift are on the left side).

Figure C1 .
Figure C1.Luminosity function at  ∼ 9 for the models described in the main text.The -axis shows the rest frame UV absolute magnitude.We compare our results with a compilation of results from JWST(Leung et al. 2023;Harikane et al. 2023;Finkelstein et al. 2023), including lensed fields(Willott et al. 2024).

Table 1 .
Fiducial *  * log 10 (  /SFR) values in our Fisher forecast; for the FFB-related parameter  max we consider two cases, as discussed in Sec.2.2.2.Other cosmological and astrophysical parameters in 21cmFAST are fixed throughout the work.